When the system is described by a mixed state, another important quantity to study is the entanglement of purification which measures the total (quantum and classical) correlation between two disjoint subsystems. In order to define the EoP, we consider a mixed bipartite system with density matrix \(\rho _{AB}\) . we can always purify this mixed state into a pure state \(\vert \psi _{AA^{\prime }BB^{\prime }}\rangle \)  by adding auxiliary degrees of freedom to the Hilbert space as \(\mathcal {H}_A\otimes \mathcal {H}_B\rightarrow \mathcal {H}_A\otimes \mathcal {H}_B \otimes \mathcal {H}_{A^{\prime }}\otimes \mathcal {H}_{B^{\prime }}\)  such that the total density matrix in enlarged Hilbert space is given by \(\rho _{AA^{\prime }BB^{\prime }}=\vert \psi _{AA^{\prime }BB^{\prime }}\rangle \langle \psi _{AA^{\prime }BB^{\prime }}\vert \) . This pure state is called a purification of \(\rho _{AB}\)  if we have
\(\rho _{AB}=Tr_{A^{\prime }B^{\prime }}\left(\vert \psi _{AA^{\prime }BB^{\prime }}\rangle \langle \psi _{AA^{\prime }BB^{\prime }}\vert \right)\) . The EoP is then defined by minimizing the entanglement entropy \(S_{AA^{\prime }}\)  over all purifications of \(\rho _{AB}\)  [1]}
\(E_p(\rho _{AB})=\underset{\vert \psi _{AA^{\prime }BB^{\prime }}\rangle }{\rm {min}}(S_{AA^{\prime }}),\) 
